Given 
\begin{aligned} \operatorname{Area}(E F G H) & =E K \cdot F G \\ & =(E N+K N) \cdot \frac{1}{2} B D \\ & =\frac{1}{2} B D \cdot E N+\frac{1}{2} B D \cdot K N \\ & =\frac{1}{4} B D \cdot A I+\frac{1}{4} B D \cdot C J \\ & =\frac{1}{2} \operatorname{Area}(A B D)+\frac{1}{2} \operatorname{Area}(B C D) \\ & =\frac{1}{2} \operatorname{Area}(A B C D) . \end{aligned}
How do you know $EN=\dfrac{AI}{2}$ and $KN=\dfrac{CJ}{2}$ and $\dfrac{Area(ABD)}{2}+\dfrac{Area(BCD)}{2}=\dfrac{Area(ABCD)}{2}$?